Figure 1.
Comparison of node affinity to sub-graph affinity.
This figure illustrates a pixel-to-pixel rectangular sub-graph specific to implementation for an image and the corresponding edge weight equations. (a) For node affinity, each node, , (red) is compared to a neighbouring node,
(green), for all nodes in the local neighborhood,
. (b) If neighborhood statistics, such as a mean or weighted average, were applied as preprocessing to the data or image, the resulting similarity metric would contain regional statistics and would be more robust to noise with the cost of losing fine details such as edges and textures. (c) Rather than calculating the similarity of two regions from a single statistic, as in (b), sub-graph affinity calculates the similarity of two regions by calculating the similarity of each corresponding element within the regions, improving robustness to noise while preserving fine details of the region. Using sub-graph affinity, two regions with the same average intensity, but with differing textures, can be classified as two different classes.
Figure 2.
Effects of graph kernel size and sub-graph kernel size on neighbourhoods and algorithm complexity.
The effects of increasing the size of the graph kernel, , and increasing the size of the sub-graph kernel,
, are illustrated using a two-dimensional example above. As the size of
increases, from top to bottom, the size of the neighbourhood (solid dark blue) surrounding
(dark red) increases. As the size of
increases, from left to right, the size of the sub-graphs (light red and light green) increases. Increasing the size of
decreases the sparsity of W and increasing the size of
increases the computational complexity of calculating
, but only moderately.
Figure 3.
Overview of spectral clustering algorithm with statistical sub-graph affinity.
The image is lexicographically unwrap into a vector, spatially weighted kernels and
are constructed, edge weights are calculated, and traditional spectral clustering techniques are applied before the labelled data is wrapped back into a labelled image. Although the kernels are illustrated as matrices above, the kernels can be constructed as an arbitrary graph.
Figure 4.
Real-world images used for testing.
The performances of spectral clustering using node affinity and sub-graph affinity were evaluated using the above test images, reproduced from the Berkeley Segmentation Dataset [23].
Figure 5.
Segmentation results show how well the sub-graph affinity model outperforms several node affinity models.
For all tests, the proposed statistical sub-graph affinity model performs as well as or outperforms the node affinity model. The dotted lines represent the mean f1-measure for the corresponding nodal affinity model. The solid lines represent the mean f1-measure for the statistical sub-graph affinity models associated with sub-graph kernel sizes of 3×3 through 9×9. The proposed sub-graph affinity model outperforms existing nodal affinity models at low PSNR.
Table 1.
The effect of noise on spectral clustering f1-measures is illustrated.
Figure 6.
Object boundaries resulting from sub-graph a_nity and node a_nity segmentation compared to ground truth.
The object boundaries resulting from the segmentation process are illustrated in red; the ground truth boundaries are illustrated in green. In the presence of noise, spectral clustering using the proposed statistical sub-graph a_nity model produces more robust results than the other tested methods which use the node a_nity model. Noise tolerance is most evident in the ‘owls’ image, where the segmentation results produced by the proposed statistical sub-graph a_nity model consists of large homogeneous segments, whereas that produced by the other tested methods using a node a_nity model contain many small noisy segments. Furthermore, the proposed statistical sub-graph a_nity model better handles the presence of textural characteristics within the images, as evident in the ‘tiger’ image; the body of the tiger is identi_ed as a single segment by the proposed model.